may want to check the coil
mbayer,
Not clear from your post as to whether or not you had the carb properly tuned. I think the process is new air and fuel filter, new plug properly gapped, then have the carb set by a dealer.
Dumb question, but is your fuel fresh? Mixed in the last 30 days fresh? And do you shake the can well before you refuel your saw to mix it well?
I think it is possible as well that you could have a bad coil. What I'd do is go run it until it dies again, then check for spark at the spark plug. Pull the plug, look at the gap in the electrode to make sure there's no carbon fouling it (which would kill the spark), and then ground the plug against the head with insulated pliers and pull the cord. You should see a bright blue spark in the plug gap. If you have this, but the saw will not start, then I think you have a fuel problem. Could be possibly getting too hot and vapor locking, but this is not common with your saw.
Some saws do get too hot and I think this can affect the coil as well, but I think if that happens you will easily know that because you'll have no spark at the plug gap.
You may want to pull the starter cover and make certain the flywheel fan is not choked up with crud and that the cooling fins on the cylinder head are clean so they will dissipate heat. Also make sure the carb heating shutter is installed to summer position. Sometimes people put this back in on the winter setting and don't realize there's a difference. Don't know if it would make much difference where you live, but it's there to prevent carb icing in cold conditions.
There are various additives that you can use to clean up a carb. Other members may have suggestions, but I'd use Sea Foam if it were mine. You measure it out and add it to the gas.
The other poster just indicated he'd put in a carb kit. I imagine his carb was cleaned at the same time.
You only need fuel, compression, and ignition to make this saw run. If it's running, you have compression. If you can see the spark right after it shuts down, you've got spark. That leaves a fuel problem of some kind.
Good luck. There are far better experts than me on this saw lurking on AS, maybe they'll see your thread and offer better advice.